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The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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Look Before You Leap

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

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Two Ladders

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

Take a Square II

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

You may wish to look at Take a Square before trying this problem.
Diagonal and fold to midpoint
Diagonal and fold to one eighth
You can divide the long diagonal of a square into different fractions by folding.
  • In the first image the second fold joins a corner of the square to the midpoint of the opposite side.
  • In the second image the second fold joins a corner of the square to a point $\frac{1}{8}$ of the way along the opposite side.

This problem is about the fractions of the long diagonal of a square which you can construct in this way.

To start with, we shall only consider points on the side of the square which can easily be found by folding. That is, $\frac{1}{2}$s or $\frac{1}{4}$s or $\frac{1}{8}$s and so on.
Investigate the fractions of the long diagonal of a square that can be created in the way described above. Here are some examples to think about:

Quarter1Quarter2Quarter3Quarter4...All quarters


Eighth1Eighth2Eighth3Eighth4...All eighths

Can you extend the findings and make generalisations?
Can you justify your generalisations?

What about starting with fractions of the side of the square that are not so easily found by folding?